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Evaluation of Level Set reinitialization algorithms for phase change simulation on unstructured grids
Guillaume Sahut, Giovanni Ghigliotti, Philippe Marty, Guillaume Balarac
To cite this version:
Guillaume Sahut, Giovanni Ghigliotti, Philippe Marty, Guillaume Balarac. Evaluation of Level Set reinitialization algorithms for phase change simulation on unstructured grids. 2019. �hal-02324919�
EVALUATION D’ALGORITHMES DE REINITIALISATION DE
1
LA LEVEL SET POUR LA SIMULATION DU CHANGEMENT DE
2
PHASE SUR MAILLAGES NON STRUCTURES
3 4 5
Guillaume Sahut(1), Giovanni Ghigliotti(1)
6
Philippe Marty(1), Guillaume Balarac(1)
7
(1)Univ. Grenoble Alpes, CNRS, Grenoble INP, LEGI, 38000 Grenoble, France - e-mail: guillaume.sahut@univ-grenoble-alpes.fr 8
9
Dans cette étude, on présente différentes méthodes numériques pour la simulation du changement de phase 10
liquide-vapeur (ébullition). On utilise un formalisme Level Set pour capturer l’interface liquide-vapeur. Un tel 11
formalisme nécessite une étape de réinitialisation de la fonction Level Set après advection. Cette étape est critique 12
pour la simulation du changement de phase car elle ne doit ni déplacer l’interface, ni introduire de déformations 13
dans le profil de la fonction Level Set, sous peine de détériorer la précision du calcul de la normale à l’interface 14
et de sa courbure, nécessaires pour définir respectivement la vitesse de l’interface due au changement de phase et 15
le saut de pression à l’interface. On présente d’abord les équations résolues et le couplage des équations de Navier- 16
Stokes avec le taux de transfert de masse modélisant le changement de phase. Puis on détaille différents 17
algorithmes de réinitialisation de la Level Set pour la simulation numérique de l’ébullition, sur maillages 18
structurés et non structurés. Ces méthodes sont ensuite validées par un cas-test de croissance de bulle statique à 19
taux de transfert de masse fixé. En particulier, on observe qu’à l’instant correspondant au doublement du rayon 20
de la bulle, ce dernier converge en maillages pour toutes les méthodes présentées. Le test concluant sur maillages 21
non structurés ouvre la voie à la simulation du changement de phase liquide-vapeur dans des géométries 22
complexes.
23 24
MOTS CLEFS : écoulements diphasiques, changement de phase, maillages non structurés, Level Set,
25
ébullition
26
27
Evaluation of Level Set reinitialization algorithms for phase change
28
simulation on unstructured grids
29
In this study, we present different numerical methods for the simulation of liquid-vapor phase change (boiling).
30
We use a Level Set formalism to capture the liquid-vapor interface. Such a formalism requires a reinitialization 31
(aka redistancing) step of the Level Set function after advection. This step is critical for phase change simulation 32
as it must neither move the interface nor induce perturbations in the Level Set function, otherwise the normal 33
vector to the interface and its curvature, two quantities that are crucial to define respectively the interface velocity 34
due to phase change and the pressure jump at the interface, would be in turn too much perturbed. Here we present 35
a comparison of different reinitialization algorithms of the Level Set function for boiling simulation, on structured 36
and unstructured grids. These methods are then validated against the analytical case of a static growing bubble 37
with a fixed mass transfer rate. In particular, we observe that at the time corresponding to a doubled bubble radius, 38
the error on the bubble radius decreases with the grid cell size for all presented methods.
39
KEYWORDS: two-phase flows, phase change, unstructured grids, Level Set, boiling
40
I INTRODUCTION
41
Two-phase flows are encountered in a wide range of industrial applications such as heat exchangers, nucleate
42
boiling or spray cooling. The particularity of two-phase flows is the existence of an interface between the two
43
phases. Numerical simulations of two-phase flows require the localization of the interface. Two-phase flow
44
simulations including liquid-vapor phase change are even more challenging, as the motion of the liquid-vapor
45
interface depends also on the mass transfer rate. There exist several numerical methods to keep track of the
46
interface. In this study we describe different versions of the Level Set method [Osher and Sethian, 1988] on
47
both structured and unstructured grids. These methods are then validated and compared on the analytical case
48
of a static growing bubble with an imposed mass transfer rate.
49
II GOVERNING EQUATIONS
50
We solve the incompressible Navier-Stokes equations
51
1 u p
t u
u , (1)
52
where u
is the velocity, 𝑝 the pressure, 𝜌 the density and Σ the viscous stress tensor of the given phase. The
53
discontinuity in the velocity field at the interface is given by
54
u m 1 n,
(2)55
where
q qliq qvap denotes the discontinuity, or jump, of the quantityq
across the interface , m is56
the mass transfer rate (in kg m-2 s-1) responsible for phase change, and n
is the interface normal vector pointing
57
towards the liquid phase. When solving (1) we also have to take into account the pressure jump at the interface
58
given by
59
m2 1
p , (3)
60
where
is the surface tension and
the curvature of the interface. Both jumps (2) and (3) are used in the61
projection method to solve (1) [Tanguy et al., 2014].
62
In this study, as we focus on the different Level Set approaches available to accurately capture the interface,
63
we assume that the mass transfer rate is fixed. These methods have been implemented in the YALES2 solver
64
[Moureau et al., 2011] and are presented in the next section.
65
III LEVEL SET METHODS ON STRUCTURED GRIDS
66
The most challenging task in two-phase flow simulations is the accurate localization and advection of the
67
interface at every time step. Perturbations in interface localization result in a loss of precision and, in the more
68
severe cases, in physical aberrations. With phase change, this well-known problem is even more restrictive.
69
There are several methods to represent the interface. In this work, the Level Set method is used. A Level Set
70
function is a function seen as a set of iso-levels. The liquid-vapor interface is identified as one specific iso-
71
level [Osher and Sethian, 1988]. The Level Set method has two major advantages. First, the advection of the
72
Level Set function in all the computational domain enables the implicit capture of the interface. Second,
73
geometrical properties such as normal vector and curvature of the interface are embedded in the Level Set
74
field. The main drawback of the Level Set method is the mandatory need of a reinitialization step of the Level
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Set function.
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III.1 Case 1 - Signed Distance Function reinitialized by a Hamilton-Jacobi equation
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We start our investigations by the method described by Tanguy et al. [2014]. In this method, the Level Set
78
function is the Signed Distance Function to the interface defined for any x
in the computational domain
79
by80
x x x x
min
)
( . (4)
81
In this case, the interface is identified as the 0 iso-level of the Level Set function [Osher and Sethian, 1988].
82
The Signed Distance Function is advected by solving the standard advection equation
83
,
vap vap
u m
t
(5)
84
where uvap is the vapor phase velocity, vap is the vapor density and the source term on the right hand side is
85
due to phase change. After advection, the function is reinitialized as a Signed Distance Function by solving
86
the Hamilton-Jacobi PDE
87
0
1
0,
S (6)
88
where
is a pseudo-time, S is a smoothed sign function defined by Sussman et al. [1994], and 0 is the89
previously advected Level Set that needs to be reinitialized. The equation (6) is solved in pseudo-time until
90
convergence, i.e. until
1, which is part of the definition of the Signed Distance Function. The normal91
vector to the interface n
and the curvature of the interface
are then given by92
n and n.
(7)93
Note that high precision is needed in the reinitialization of to avoid perturbations in the normal vector and
94
curvature of the interface. To this purpose, the term
in (6) is computed using a high-order scheme such as
95
the Fifth Order WENO scheme [Jiang and Peng, 2000]. High-order schemes require higher computational cost,
96
are difficult to implement on unstructured grids and may reduce performance in a parallel code.
97
III.2 Case 2 - Signed Distance Function reinitialized by the Fast Marching Method
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Another method for the reinitialization of the Signed Distance Function is the Fast Marching Method
99
[Sethian, 1996]. The Fast Marching Method, based on the solution of an Eikonal equation, can be seen as the
100
stationary version of the Hamilton-Jacobi equation (6) and is given by
101
.
1
(8)102
The solution of equation (8) is based on the propagation of the Signed Distance Function values from the
103
interface along the normal direction to the interface. To limit the computational time needed to perform the
104
Fast Marching Method, we solve it only on a narrow band around the interface, large enough to be able to
105
compute the normal vector and the curvature by equations (7).
106
III.3 Case 3 - Conservative Level Set on structured grids
107
In order to improve mass conservation in each phase, Olsson and Kreiss [2005] proposed the Conservative
108
Level Set method. In this method, the Level Set function
is a smeared-out Heaviside function defined for109
x
by
110
2 ( )
) tanh ( 2 1 2 ) 1
( x
x x
, (9)111
where is the Signed Distance Function and
is a scale parameter roughly of the order of the grid cell size.112
In this case, the interface is identified as the 0.5 iso-level of the Level Set function. The Conservative Level
113
Set is advected by solving the conservative form of the standard advection equation with source term
114
uvap
uPC uvap,t
(10)
115
where the source term is composed of the phase change contribution and the divergence of the vapor velocity,
116
which should be null, but it is recommended to account for this correction to decrease the sensitivity to errors
117
in the evaluation of the velocity field. In (10), the interface velocity due to phase change uPC
is given by
118
, m n u
vap PC
and the gradient of the Level Set
is computed as
119
, ) ( 2
) cosh (
) ( 4 ) 1 (
2
n x x x
x
(11)
120
where is again the Signed Distance Function. This technique to compute
is reused from the
121
reinitialization step suggested by Chiodi and Desjardins [2017] and presented below. The Conservative Level
122
Set embeds the interesting property of volume conservation. The normal vector and the curvature of the
123
interface are given by a similar approach as in (7).
124
The reinitialization equation for the Conservative Level Set proposed by Olsson and Kreiss [2005] is
125
1 n
n
n
,
(12)126
where the normal vector is given by
127
n . (13)
128
Nevertheless, the necessity to set
to a small value to improve volume conservation produces sharp gradients129
in
and thus potential oscillations in n. To avoid this problem, and with the aim of increasing accuracy,
130
Chiodi and Desjardins [2017] built a new reinitialization equation for the Conservative Level Set given by
131
map n n
map
1 cosh 2
4 1
2
, (14)
132
where the inverse of the Conservative Level Set map is given by
133
map log 1 . (15)
134
The normal vector is computed based on the Signed Distance Function, derived from the Conservative Level
135
Set function by using the Fast Marching Method:
136
FMM
n FMM
, (16)
137
where
FMM is the Signed Distance Function given by the Fast Marching Method using boundary conditions138
on the closest nodes to the interface.
139
IV LEVEL SET METHODS ON UNSTRUCTURED GRIDS
140
IV.1 Case 4 - Signed Distance Function on unstructured grids
141
To be able to address complex geometries, algorithms are now extended on unstructured grids. The first
142
unstructured case is similar to Case 1. Only the reinitialization step of the Signed Distance Function is replaced
143
by the method developed on unstructured grids by Dapogny and Frey [2012]. We make use of the MshDist
144
library implementing this method.
145
IV.2 Case 5 - Conservative Level Set on unstructured grids
146
Here we want to extend the methodology of Chiodi and Desjardins [2017] described for structured grids in
147
Section III.3 to unstructured grids. The only difference is the replacement of
FMM by the Signed Distance148
Function given by the method proposed by Dapogny and Frey [2012].
149
V NUMERICAL RESULTS
150
We validate the different Level Set reinitialization methods presented in the previous section on the case of
151
a 2D static growing bubble with a fixed mass transfer rate from [Tanguy et al., 2014]. The initial bubble radius
152
is R0 = 10-3 m and the imposed mass transfer rate is m = 10-1 kg m-2 s-1. The simulations are performed until
153
final time tf = 10-2 s needed for the bubble radius to double the initial radius. The other physical parameters
154
of interest are liq = 103 kg m-3, vap = 1 kg m-3,
= 7 10-2 N m-1, liq = 10-3 kg m-1 s-1 and vap= 1.78155
10-5 kg m-1 s-1. In Table 1, the methods used in all cases are summarized.
156
157
Grid topology Structured Unstructured
Level Set type SDF CLS SDF CLS
Reinitialization
method HJ eq. FMM [Chiodi and
Desjardins, 2017]
[Dapogny and Frey, 2012]
[Dapogny and Frey, 2012] + [Chiodi and Desjardins, 2017]
Cases Case 1 Case 2 Case 3 Case 4 Case 5
158
Table 1. Summary of all cases presented in the previous sections. SDF stands for Signed Distance Function, CLS for 159
Conservative Level Set, HJ for Hamilton-Jacobi and FMM for Fast Marching Method.
160 161
The five cases have been computed on four different grid cell sizes. As an example, Fig. 1 shows the results
162
of Case 5 at final time on four different unstructured grids. The relative error on the bubble radius with respect
163
to the theoretical radius at final time is given for all cases at final time in Fig. 2 (a). One can see that on the
164
finest grid, the Case 5 detailed in Fig. 1 presents the highest relative error. The five methods have a convergence
165
rate close to one. For the finest grid, all relative errors on the bubble radius are below 1%. The relative errors
166
on the normal vector and the curvature are shown in respectively Fig. 2 (b) and 2 (c). The results show that the
167
error on the normal vector decreases at order 1 for all methods including the Signed Distance Function on
168
unstructured grids (Case 4). For Case 5, i.e. for the Conservative Level Set on unstructured grids, further work
169
is needed to improve the convergence of both normal vector and curvature.
170
(a) (b) (c) (d)
Fig. 1. The results for the Case 5 on four different unstructured grids are plotted at final time with a characteristic cell size of (a) 4 10-4 m, (b) 2 10-4 m, (c) 1 10-4 m and (d) 5 10-5 m. The initial interface is plotted in blue, the computed interface in black, and the theoretical interface in red. For clarity, only the coarsest grid is represented in (a).
The computed liquid and vapor velocity fields are plotted in (b).
171 172
(a) (b) (c)
Fig 2. The normalized L∞ norm of the error on the bubble radius is plotted at final time for the five cases on four different grid cell sizes (a). The analogous error for the normal vector (b) and for the curvature (c).
VI CONCLUSION AND PERSPECTIVES
173
In this study, several numerical methods for the advection and reinitialization of the Level Set function have
174
been presented in the context of liquid-vapor phase change. These methods have been validated and compared
175
on the case of a static growing bubble with a fixed mass transfer rate. All methods present a convergence rate
176
with the grid cell size close to one. Two of the five methods presented are designed for unstructured grids. The
177
ability to accurately compute the interface position allows the quantitative simulation of liquid-vapor phase
178
change on unstructured grids. This opens the path to numerical simulations of liquid-vapor phase change on
179
complex geometries.
180
The main perspective of our work is the simulation of phase change on unstructured grids with a computed
181
mass transfer rate which depends on the thermal fluxes at the interface and on the latent heat of the fluid.
182
VII ACKNOWLEDGMENTS
183
This work has been supported by the LabEx Tec21 (Investissements d’Avenir-Grant Agreement No. ANR-
184
11-LABX-0030). Vincent Moureau and Ghislain Lartigue from the CORIA lab and the SUCCESS scientific
185
group are acknowledged for providing the YALES2 code. G. B. is also grateful for the support from Institut
186
Universitaire de France. The authors would like to thank Patrick Begou for support on algorithms
187
implementation and Charles Dapogny for support on the MshDist library.
188 189
VIII REFERENCES
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Chiodi R., Desjardins O. (2017). — A reformulation of the conservative level set reinitialization equation for
191
accurate and robust simulation of complex multiphase flows. J. Comput. Phys., 343: 186-200.
192
Dapogny C., Frey P. (2012). — Computation of the signed distance function to a discrete contour on adapted
193
triangulation. Calcolo, 49(3): 193-219.
194
Jiang G.-S., Peng D. (2000). — Weighted ENO Schemes for Hamilton-Jacobi Equations. SIAM J. Sci.
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Comput., 21(6): 2126-2143.
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Moureau V., Domingo P., Vervisch L. (2011). — Design of a massively parallel CFD code for complex
197
geometries. CR Mec, 339(2-3): 141-148.
198
Olsson E., Kreiss G. (2005). — A conservative level set method for two phase flow. J. Comput. Phys., 210:
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Osher S., Sethian J. A. (1988). — Fronts propagating with curvature-dependent speed: Algorithms based on
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Hamilton-Jacobi formulations. J. Comput. Phys., 79: 12-49.
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Sethian J. A. (1996). — A Fast Marching Level Set Method for Monotonically Advancing Fronts. Proc. Natl.
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Acad. Sci., 93(4): 1591-1595.
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Sussman M., Smereka P., Osher S. (1994). — A Level Set Approach for Computing Solutions to
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Incompressible Two-Phase Flow. J. Comput. Phys., 114: 146-159.
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Tanguy S., Sagan M., Lalanne B., Couderc F., Colin C. (2014). — Benchmarks and numerical methods for the
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